On the number of solutions of two-variable diagonal quartic equations over finite fields

Let $p$ be a odd prime number and let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements. In this paper, by using the Gauss sum and Jacobi sum, we give an explicit formula for the number $N(x_1^4+x_2^4=c)$ of solutions of the following two-variable diagonal quartic equations...

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Bibliographic Details
Published inAIMS mathematics Vol. 5; no. 4; pp. 2979 - 2991
Main Authors Zhao, Junyong, Zhao, Yang, Niu, Yujun
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2020
Subjects
diagonal hypersurface
finite field
gauss sum
jacobi sum
rational point
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Summary:Let $p$ be a odd prime number and let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements. In this paper, by using the Gauss sum and Jacobi sum, we give an explicit formula for the number $N(x_1^4+x_2^4=c)$ of solutions of the following two-variable diagonal quartic equations over $\mathbb{F}_q$: $x_1^4+x_2^4=c$ with $c\in\mathbb{F}_q^*$. From this result, one can deduce that $N(x_1^4+x_2^4=c)=q+O(q^{\frac{1}{2}}).$
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2020192